My two cents: this does not sound like a GIO problem. Their work e-mail server is far more likely the problem due to the weird "bouncing" or "greylisting" events so I would tell them to use another email address due to "an apparent incompatibility with list services."
It also sounds like they could have been hacked since leaving and re-joining is happening via the web so I would most strongly urge them to change their work email password and their GIO account password after they have changed to a new email address.
If their computer isn't fully compromised, I would bet it would would resolve their issues. If it continues, their computer is likely infected.